On the Power of Recursive Word-Functions Without Concatenation

Abstract

Primitive recursion can be defined on words instead of natural numbers. Up to usual encoding, primitive recursive functions coincide. Working with words allows to address words directly and not through some integer encoding (of exponential size). Considering alphabets with at least two symbols allows to relate simply and naturally to complexity theory. Indeed, the polynomial-time complexity class (as well as NP and exponential time) corresponds to delayed and dynamical evaluation with a polynomial bound on the size of the trace of the computation as a direct acyclic graph.

Primitive recursion in the absence of concatenation (or successor for numbers) is investigated. Since only suffixes of an input can be output, computation is very limited; e.g. pairing and unary encoding are impossible. Yet non-trivial relations and languages can be decided. Some algebraic ($\texttt {a}^n\texttt {b}^n$, palindromes) and non-algebraic ($\texttt {a}^n\texttt {b}^n\texttt {c}^n$) languages are decidable. It is also possible to check arithmetical constrains like $\texttt {a}^n\texttt {b}^m\texttt {c}^{P(n,m)}$ with P polynomial with positive coefficients in two (or more) variables. Every regular language is decidable if recursion can be defined on multiple functions at once.